Solvability of nonlinear elliptic equations with gradient terms

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on Δu±g(|∇u|)=f(u), u⩾0, where f and g are increasing continuous functions. We give conditions on f and g which guarantee the availability or the absence of positive solutions of such equations in RN. Our results considerably improve the existing ones and are sharp or close to sharp in the model cases. In particular, we completely characterize the solvability of such equations when f and g have power growth at infinity. We also derive a solvability statement for coercive equations in general form.